Review

GRE Data Analysis - Counting Methods

There are many techniques that you can use to help you better understand a data set. The simplest of these are the concepts of mean and average. However, that's just the beginning. On the GRE revised General Test, you can expect to see many types of math
questions, some of which will require you to rely on counting methods that are more advanced than just the mean or average. Several of the counting methods that appear on the GRE exam are presented in this section.

The Multiplication Principle

The concept of multiplication is one of the more simplest math counting methods that you can expect to see on the GRE revised General Test. The Multiplication Principle can be stated as following:

If there are k choices for one thing and m choices for another thing, then there are k × m or km different possibilities for the pair of choices.

Practice Question

An ice cream parlor sells three flavors of ice cream: vanilla, chocolate and strawberry, and offers two choices of topping: chocolate syrup and sprinkles.

Question: A group of friends visit the ice cream parlor, but are only allowed one flavor of ice cream and one topping. How many different possibilities are there?

Solution: You can enumerate possible combinations:

Amber likes vanilla ice cream with chocolate syrup.

Ben likes vanilla ice cream with sprinkles.

Clarisse likes chocolate ice cream with chocolate syrup.

David likes chocolate ice cream with sprinkles.

Enrico likes strawberry ice cream with chocolate syrup.

Fiona likes strawberry ice cream with sprinkles.

Gus likes...

For Gus, there are no more unique possibilities, so he has to have the same combination as one of his friends. Thus, the total number of possibilities is:

3 × 2 = 6

In other words, the number of choices of ice cream and toppings is equal to the number of flavors of ice cream times the number of toppings.

The Multiplication Principle applies as long as the choice of one thing is independent of the choice of the other thing. If, for example, the owner of the ice cream parlor refuses to serve chocolate ice cream with chocolate syrup, then Clarisse is out
of luck. The number of pairs of choices would be reduced to 5 and the Multiplication Principle would no longer apply.

The Multiplication Principle applies also when there are many choices of many things, as long as they are all independent. So, if Joe has 5 shirts, 7 pants, 4 jackets and 6 pairs of shoes (and every choice is allowed), his total number of outfits would
be

5 × 7 × 4 × 6 = 840

The Multiplication Principle can be applied to probability experiments where there are two possible outcomes at each trial and the trials are independent. For example if a coin is tossed seven times, then there are two possible outcomes (Heads or Tails)
at each trial. The total number of possible outcomes is:

2 × 2 × 2 × 2 × 2 × 2 × 2 = 27 or 128

Factorials

For the set of Natural numbers {1, 2, 3, 4, 5, ...}, the factorial of n, if n is a natural number, is defined to be the product of all the natural numbers less than or equal to n, and is denoted by n!

For example:

4! = 4 × 3 × 2 × 1 = 247! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

Factorials are used in the calculation of Permutations and Combinations.

Permutations and Combinations

The concept of permutations and combinations is concerned with counting and forming subsets from a larger set. The simpler permutation and counting questions on the GRE revised General Test can be answered with pen and pencil, and by enumerating all possible
combinations. But the more difficult questions will require you to know and use a few standard formula. For example, considering the following question:

There are 8 tracks on your favorite album, but you only have time to listen to 3 of them. How many unique choices of 3 tracks do you have?

Let’s label the tracks A, B, C, D, E, F, G and H. The first track you choose could be A, B, C, D, E, F, G or H, so there are 8 choices. The second track you choose could be any one of the seven remaining tracks, so there are 7 choices to choose the second
track. The third track you choose could be any one of the six remaining tracks, so there are 6 choices for the third track. So, applying the Multiplication Principle, altogether for this example, you have:

8 × 7 × 6 = 336 choices

We say there are 336 arrangements, or Permutations, of 3 objects chosen from 8. But are all of these choices different? That depends, and that is where some of the more difficult GRE math questions about permutations come
in.

Suppose on Monday you choose track A first, followed by track B, followed by track C (ABC) then on Tuesday you choose track B first, followed by track A, followed by track C (BAC). Are these different or the same? Is ABC the same as BAC?

It depends. If A, B and C are your three absolute favorites, then it probably doesn’t matter which order you play them in. The choice is the same, but the order is different. However, if you like playing track A while you’re brushing your hair, track
B while you’re cleaning your teeth and track C while you’re putting on your shoes, then order might matter. But probably you don’t really care about the order as long as you get to play your three favorites. If order doesn’t matter, then
the six orderings are:

ABC, ACB, BAC, BCA, CAB and CBA

All these give you the same listening experience, so the number of choices is 336 ÷ 6 = 56.

We say there are 56 choices, or Combinations, of 3 objects chosen from 8. So, for Combinations order doesn't matter but for Permutations it does. The two important ideas:

Combinations give the number of choices without regard to order.

Permutations give the number of arrangements.

If you want to select k objects (or arrangements) from n choices, then the number of permutations is given by the following formula:

And the number of combinations is:

Another way to denote permutations is by

nPk

And, the number of combinations can be written as:

nCk

In all these formula, n is the number of choices, and k is how many of the choices are selected. When confronted with a counting or permutation question, you have to decide if order matters. If order matters, then use the formula for permutations, else
use the formula for combinations. Below are two examples.

Practice Question

Question: A committee of 4 people is to be chosen from 9 candidates. How many ways can you do this?

Solution: Here, n=9, and k=4, thus:

If the order does matter:

Practice Question

Question: A committee of 4 people is to be chosen from 9 candidates. The first person you choose is to be the President, the second person you choose is to be the Secretary, and the third person you choose is to be
the Treasurer. In how many ways can you do this?

Solution: Here, n=9, and k=4, just as before, but because order does matter, you have to use the permutation formula: